|
|
||
|
202 iX ANALYTIC FUNCTIONS
|
||
|
|
||
|
gk(u) = £ 6<*V, Gjfe(w) = £ I*?' !«*• On the other hand' let (av^v) be a power
# /i
series In p variables with coefficients in E, which is absolutely summable
in a polydisk P of Kp, of center 0 and radii rk(l^k^p). If, in a monomial zv = znl1 ...zn/, we replace "formally" each zk by the power series gk(u\ we are led to take the formal "product" of n^ + n2 + • • • + np series, i.e. to pick a term in each of the n± + • - • + np factors, to take their product and then to "sum" all terms thus obtained. We are thus led to consider, for each v = (X, n2 , . . . , np) the set Av of all finite families Qj,kJ) = p where fikj e N*, k ranges from 1 to p, and for each k, j ranges from 1 to nk\ to such a p we associate the element |
||
|
|
||
|
nk
|
||
|
|
||
|
With these notations:
(9.2.1) Suppose sl9...,sq are q numbers >0 satisfying the conditions
Gjtfo,..., sq) < rk for l^k^p. Then, for each u in the open polydisk S c Kq of center 0 and radii st (1 < i" < q), the family (tp(u)) (where p ranges through the denumerable set of indices A = (j Av) is absolutely summable,
and iff(z) = £ avzv, its sum is equal toftg^u), g2(u), ..., gp(u)).
|
||
|
|
||
|
In other words, under the conditions Gk(sl9 ..., sq) < rk (l^
"substitution" of the series gk(u) for zk (1 < k </?) in the series / yields an absolutely summable family, even before all the terms tp(u) having the same degrees in ul9..., uq have been gathered together.
To prove (9.2.1), we need only prove that the family (tp(u)) is absolutely
summable; that its sum is/(^(w)?..., gp(u)) follows by application of the associativity theorem (5.3.6) to the subsets Av of A, and by using (5.5.3), which shows that £ tp(u) is equal to a^g^u))"1 ... (gp(u))np. To prove the
peAv
family (tp(u)) (peA) is absolutely summable, we apply (5.3.4). For any
finite subset B of A, we have, by (5.3.5) and (5.5.3)
Z uncoil < KB • (Gi(s19..., sjr... (QP(sl9..., s,))"'
p 6 B n Av
and by assumption, the right-hand side of that inequality is the element of
index v of an absolutely summable family; hence the result.
Write tp(u) = cpu\ with A = (^,..., ^), ^ = f V mkji (if we have
*=i j~i
Hkj = (w*ji» • • •»^> ))• From (9.2.1) and (5.3.5) it follows (taking all the ut |
||
|
|
||